I am trying to understand the above proof. My problem is that I fail to understand why we need to construct the neighborhood $W_1\cap...\cap W_n\cap H_x$ of $x$ that interescts with finite many of $\mathcal V$. Because both $W_1\cap...\cap W_n$ and $H_x$ seem to be neighborhoods of $x$ that interescts with finite many of $\mathcal V$.
I don't see why $H_x \in \mathcal{H}_n$ would intersect only finitely many members of $\mathcal{V}$. It does not intersect any $V_H$ for $H$ with higher minimal index and we want to preserve that. But we also have to take care of the first $n$ many $\mathcal{H}_n$, so taking a $W_i$ for each $i$ smaller or equal takes care of that and intersecting with $H_x$ (making it smaller) adds the property we don't intersect larger indexed $V_H$.. Neither is enough by itself but the intersection achieves both, similar to taking the minimum of some finitely many $\delta$'s in some metric continuity proofs.